The Hindu-Arabic Numerals

THE

HINDU-ARABIC NUMERALS

BY
DAVID EUGENE SMITH
AND
LOUIS CHARLES KARPINSKI

BOSTON AND LONDON
GINN AND COMPANY, PUBLISHERS
1911

COPYRIGHT, 1911, BY DAVID EUGENE SMITH
AND LOUIS CHARLES KARPINSKI
ALL RIGHTS RESERVED
811.7

The Athenæum Press
GINN AND COMPANY · PROPRIETORS
BOSTON · U.S.A.

PREFACE

So familiar are we with the numerals that bear the misleading name of Arabic, and so extensive is their use in Europe and the Americas, that it is difficult for us to realize that their general acceptance in the transactions of commerce is a matter of only the last four centuries, and that they are unknown to a very large part of the human race to-day. It seems strange that such a labor-saving device should have struggled for nearly a thousand years after its system of place value was perfected before it replaced such crude notations as the one that the Roman conqueror made substantially universal in Europe. Such, however, is the case, and there is probably no one who has not at least some slight passing interest in the story of this struggle. To the mathematician and the student of civilization the interest is generally a deep one; to the teacher of the elements of knowledge the interest may be less marked, but nevertheless it is real; and even the business man who makes daily use of the curious symbols by which we express the numbers of commerce, cannot fail to have some appreciation for the story of the rise and progress of these tools of his trade.

This story has often been told in part, but it is a long time since any effort has been made to bring together the fragmentary narrations and to set forth the general problem of the origin and development of these numerals. In this little work we have attempted to state the history of these forms in small compass, to place before the student materials for the investigation of the problems involved, and to express as clearly as possible the results of the labors of scholars who have studied the subject in different parts of the world. We have had no theory to exploit, for the history of mathematics has seen too much of this tendency already, but as far as possible we have weighed the testimony and have set forth what seem to be the reasonable conclusions from the evidence at hand.

To facilitate the work of students an index has been prepared which we hope may be serviceable. In this the names of authors appear only when some use has been made of their opinions or when their works are first mentioned in full in a footnote.

If this work shall show more clearly the value of our number system, and shall make the study of mathematics seem more real to the teacher and student, and shall offer material for interesting some pupil more fully in his work with numbers, the authors will feel that the considerable labor involved in its preparation has not been in vain.

We desire to acknowledge our especial indebtedness to Professor Alexander Ziwet for reading all the proof, as well as for the digest of a Russian work, to Professor Clarence L. Meader for Sanskrit transliterations, and to Mr. Steven T. Byington for Arabic transliterations and the scheme of pronunciation of Oriental names, and also our indebtedness to other scholars in Oriental learning for information.

DAVID EUGENE SMITH

LOUIS CHARLES KARPINSKI

CONTENTS

PRONUNCIATION OF ORIENTAL NAMES

(S) = in Sanskrit names and words; (A) = in Arabic names and words.

b, d, f, g, h, j, l, m, n, p, sh (A), t, th (A), v, w, x, z, as in English.

a, (S) like u in but: thus pandit, pronounced pundit. (A) like a in ask or in man. ā, as in father.

c, (S) like ch in church (Italian c in cento).

ḍ, ṇ, ṣ, ṭ, (S) d, n, sh, t, made with the tip of the tongue turned up and back into the dome of the palate. ḍ, ṣ, ṭ, ẓ, (A) d, s, t, z, made with the tongue spread so that the sounds are produced largely against the side teeth. Europeans commonly pronounce ḍ, ṇ, ṣ, ṭ, ẓ, both (S) and (A), as simple d, n, sh (S) or s (A), t, z. ḏ (A), like th in this.

e, (S) as in they. (A) as in bed.

ġ, (A) a voiced consonant formed below the vocal cords; its sound is compared by some to a g, by others to a guttural r; in Arabic words adopted into English it is represented by gh (e.g. ghoul), less often r (e.g. razzia).

h preceded by b, c, t, ṭ, etc. does not form a single sound with these letters, but is a more or less distinct h sound following them; cf. the sounds in abhor, boathook, etc., or, more accurately for (S), the "bhoys" etc. of Irish brogue. h (A) retains its consonant sound at the end of a word. ḥ, (A) an unvoiced consonant formed below the vocal cords; its sound is sometimes compared to German hard ch, and may be represented by an h as strong as possible. In Arabic words adopted into English it is represented by h, e.g. in sahib, hakeem. ḥ (S) is final consonant h, like final h (A).

i, as in pin. ī, as in pique.

k, as in kick.

kh, (A) the hard ch of Scotch loch, German ach, especially of German as pronounced by the Swiss.

ṁ, ṅ, (S) like French final m or n, nasalizing the preceding vowel.

ṇ, see ḍ. ñ, like ng in singing.

o, (S) as in so. (A) as in obey.

q, (A) like k (or c) in cook; further back in the mouth than in kick.

r, (S) English r, smooth and untrilled. (A) stronger. ṛ, (S) r used as vowel, as in apron when pronounced aprn and not apern; modern Hindus say ri, hence our amrita, Krishna, for a-mṛta, Kṛṣṇa.

s, as in same. ṣ, see ḍ. ś, (S) English sh (German sch).

ṭ, see ḍ.

u, as in put. ū, as in rule.

y, as in you.

ẓ, see ḍ.

‛, (A) a sound kindred to the spiritus lenis (that is, to our ears, the mere distinct separation of a vowel from the preceding sound, as at the beginning of a word in German) and to ḥ. The ‛ is a very distinct sound in Arabic, but is more nearly represented by the spiritus lenis than by any sound that we can produce without much special training. That is, it should be treated as silent, but the sounds that precede and follow it should not run together. In Arabic words adopted into English it is treated as silent, e.g. in Arab, amber, Caaba (‛Arab, ‛anbar, ka‛abah).

(A) A final long vowel is shortened before al ('l) or ibn (whose i is then silent).

Accent: (S) as if Latin; in determining the place of the accent ṁ and ṅ count as consonants, but h after another consonant does not. (A), on the last syllable that contains a long vowel or a vowel followed by two consonants, except that a final long vowel is not ordinarily accented; if there is no long vowel nor two consecutive consonants, the accent falls on the first syllable. The words al and ibn are never accented.

THE HINDU-ARABIC NUMERALS

CHAPTER I

EARLY IDEAS OF THEIR ORIGIN

It has long been recognized that the common numerals used in daily life are of comparatively recent origin. The number of systems of notation employed before the Christian era was about the same as the number of written languages, and in some cases a single language had several systems. The Egyptians, for example, had three systems of writing, with a numerical notation for each; the Greeks had two well-defined sets of numerals, and the Roman symbols for number changed more or less from century to century. Even to-day the number of methods of expressing numerical concepts is much greater than one would believe before making a study of the subject, for the idea that our common numerals are universal is far from being correct. It will be well, then, to think of the numerals that we still commonly call Arabic, as only one of many systems in use just before the Christian era. As it then existed the system was no better than many others, it was of late origin, it contained no zero, it was cumbersome and little used, and it had no particular promise. Not until centuries later did the system have any standing in the world of business and science; and had the place value which now characterizes it, and which requires a zero, been worked out in Greece, we might have been using Greek numerals to-day instead of the ones with which we are familiar.

Of the first number forms that the world used this is not the place to speak. Many of them are interesting, but none had much scientific value. In Europe the invention of notation was generally assigned to the eastern shores of the Mediterranean until the critical period of about a century ago,—sometimes to the Hebrews, sometimes to the Egyptians, but more often to the early trading Phœnicians.^[1]

The idea that our common numerals are Arabic in origin is not an old one. The mediæval and Renaissance writers generally recognized them as Indian, and many of them expressly stated that they were of Hindu origin.^[2] Others argued that they were probably invented by the Chaldeans or the Jews because they increased in value from right to left, an argument that would apply quite as well to the Roman and Greek systems, or to any other. It was, indeed, to the general idea of notation that many of these writers referred, as is evident from the words of England's earliest arithmetical textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do agree, that the Chaldays, whiche fyrste inuented thys arte, did set these figures as thei set all their letters. for they wryte backwarde as you tearme it, and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and Arabike bookes ... where as the Greekes, Latines, and all nations of Europe, do wryte and reade from the lefte hand towarde the ryghte."^[3] Others, and among them such influential writers as Tartaglia^[4] in Italy and Köbel^[5] in Germany, asserted the Arabic origin of the numerals, while still others left the matter undecided^[6] or simply dismissed them as "barbaric."^[7] Of course the Arabs themselves never laid claim to the invention, always recognizing their indebtedness to the Hindus both for the numeral forms and for the distinguishing feature of place value. Foremost among these writers was the great master of the golden age of Bagdad, one of the first of the Arab writers to collect the mathematical classics of both the East and the West, preserving them and finally passing them on to awakening Europe. This man was Moḥammed the Son of Moses, from Khowārezm, or, more after the manner of the Arab, Moḥammed ibn Mūsā al-Khowārazmī,^[8] a man of great learning and one to whom the world is much indebted for its present knowledge of algebra^[9] and of arithmetic. Of him there will often be occasion to speak; and in the arithmetic which he wrote, and of which Adelhard of Bath^[10] (c. 1130) may have made the translation or paraphrase,^[11] he stated distinctly that the numerals were due to the Hindus.^[12] This is as plainly asserted by later Arab writers, even to the present day.^[13] Indeed the phrase ‛ilm hindī, "Indian science," is used by them for arithmetic, as also the adjective hindī alone.^[14]

Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn Aḥmed, Abū 'l-Rīḥān al-Bīrūnī (973-1048), who spent many years in Hindustan. He wrote a large work on India,^[15] one on ancient chronology,^[16] the "Book of the Ciphers," unfortunately lost, which treated doubtless of the Hindu art of calculating, and was the author of numerous other works. Al-Bīrūnī was a man of unusual attainments, being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in astronomy, chronology, and mathematics. In his work on India he gives detailed information concerning the language and customs of the people of that country, and states explicitly^[17] that the Hindus of his time did not use the letters of their alphabet for numerical notation, as the Arabs did. He also states that the numeral signs called aṅka^[18] had different shapes in various parts of India, as was the case with the letters. In his Chronology of Ancient Nations he gives the sum of a geometric progression and shows how, in order to avoid any possibility of error, the number may be expressed in three different systems: with Indian symbols, in sexagesimal notation, and by an alphabet system which will be touched upon later. He also speaks^[19] of "179, 876, 755, expressed in Indian ciphers," thus again attributing these forms to Hindu sources.

Preceding Al-Bīrūnī there was another Arabic writer of the tenth century, Moṭahhar ibn Ṭāhir,^[20] author of the Book of the Creation and of History, who gave as a curiosity, in Indian (Nāgarī) symbols, a large number asserted by the people of India to represent the duration of the world. Huart feels positive that in Moṭahhar's time the present Arabic symbols had not yet come into use, and that the Indian symbols, although known to scholars, were not current. Unless this were the case, neither the author nor his readers would have found anything extraordinary in the appearance of the number which he cites.

Mention should also be made of a widely-traveled student, Al-Mas‛ūdī (885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and even across the China sea, and at other times to Madagascar, Syria, and Palestine.^[21] He seems to have neglected no accessible sources of information, examining also the history of the Persians, the Hindus, and the Romans. Touching the period of the Caliphs his work entitled Meadows of Gold furnishes a most entertaining fund of information. He states^[22] that the wise men of India, assembled by the king, composed the Sindhind. Further on^[23] he states, upon the authority of the historian Moḥammed ibn ‛Alī ‛Abdī, that by order of Al-Manṣūr many works of science and astrology were translated into Arabic, notably the Sindhind (Siddhānta). Concerning the meaning and spelling of this name there is considerable diversity of opinion. Colebrooke^[24] first pointed out the connection between Siddhānta and Sindhind. He ascribes to the word the meaning "the revolving ages."^[25] Similar designations are collected by Sédillot,^[26] who inclined to the Greek origin of the sciences commonly attributed to the Hindus.^[27] Casiri,^[28] citing the Tārīkh al-ḥokamā or Chronicles of the Learned,^[29] refers to the work as the Sindum-Indum with the meaning "perpetuum æternumque." The reference^[30] in this ancient Arabic work to Al-Khowārazmī is worthy of note.

This Sindhind is the book, says Mas‛ūdī,^[31] which gives all that the Hindus know of the spheres, the stars, arithmetic,^[32] and the other branches of science. He mentions also Al-Khowārazmī and Ḥabash^[33] as translators of the tables of the Sindhind. Al-Bīrūnī^[34] refers to two other translations from a work furnished by a Hindu who came to Bagdad as a member of the political mission which Sindh sent to the caliph Al-Manṣūr, in the year of the Hejira 154 (A.D. 771).

The oldest work, in any sense complete, on the history of Arabic literature and history is the Kitāb al-Fihrist, written in the year 987 A.D., by Ibn Abī Ya‛qūb al-Nadīm. It is of fundamental importance for the history of Arabic culture. Of the ten chief divisions of the work, the seventh demands attention in this discussion for the reason that its second subdivision treats of mathematicians and astronomers.^[35]

The first of the Arabic writers mentioned is Al-Kindī (800-870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn ‛Alī, the Jew, who was converted to Islam under the caliph Al-Māmūn, is also given as the author of a work on the Hindu method of reckoning. Nevertheless, there is a possibility^[36] that some of the works ascribed to Sened ibn ‛Alī are really works of Al-Khowārazmī, whose name immediately precedes his. However, it is to be noted in this connection that Casiri^[37] also mentions the same writer as the author of a most celebrated work on arithmetic.

To Al-Ṣūfī, who died in 986 A.D., is also credited a large work on the same subject, and similar treatises by other writers are mentioned. We are therefore forced to the conclusion that the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals.

Leonard of Pisa, of whom we shall speak at length in the chapter on the Introduction of the Numerals into Europe, wrote his Liber Abbaci^[38] in 1202. In this work he refers frequently to the nine Indian figures,^[39] thus showing again the general consensus of opinion in the Middle Ages that the numerals were of Hindu origin.

Some interest also attaches to the oldest documents on arithmetic in our own language. One of the earliest treatises on algorism is a commentary^[40] on a set of verses called the Carmen de Algorismo, written by Alexander de Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 A.D. The text of the first few lines is as follows:

"This boke is called the boke of algorim or augrym after lewder use. And this boke tretys of the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this craft.... Algorisms, in the quych we use teen figurys of Inde."

CHAPTER II

EARLY HINDU FORMS WITH NO PLACE VALUE

While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek^[42] or Chinese^[43] sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along literary lines, long before the golden age of Greece. From the earliest times even up to the present day the Hindu has been wont to put his thought into rhythmic form. The first of this poetry—it well deserves this name, being also worthy from a metaphysical point of view^[44]—consists of the Vedas, hymns of praise and poems of worship, collected during the Vedic period which dates from approximately 2000 B.C. to 1400 B.C.^[45] Following this work, or possibly contemporary with it, is the Brahmanic literature, which is partly ritualistic (the Brāhmaṇas), and partly philosophical (the Upanishads). Our especial interest is in the Sūtras, versified abridgments of the ritual and of ceremonial rules, which contain considerable geometric material used in connection with altar construction, and also numerous examples of rational numbers the sum of whose squares is also a square, i.e. "Pythagorean numbers," although this was long before Pythagoras lived. Whitney^[46] places the whole of the Veda literature, including the Vedas, the Brāhmaṇas, and the Sūtras, between 1500 B.C. and 800 B.C., thus agreeing with Bürk^[47] who holds that the knowledge of the Pythagorean theorem revealed in the Sūtras goes back to the eighth century B.C.

The importance of the Sūtras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor^[48] of a Greek origin, has been repeatedly emphasized in recent literature,^[49] especially since the appearance of the important work of Von Schroeder.^[50] Further fundamental mathematical notions such as the conception of irrationals and the use of gnomons, as well as the philosophical doctrine of the transmigration of souls,—all of these having long been attributed to the Greeks,—are shown in these works to be native to India. Although this discussion does not bear directly upon the origin of our numerals, yet it is highly pertinent as showing the aptitude of the Hindu for mathematical and mental work, a fact further attested by the independent development of the drama and of epic and lyric poetry.

It should be stated definitely at the outset, however, that we are not at all sure that the most ancient forms of the numerals commonly known as Arabic had their origin in India. As will presently be seen, their forms may have been suggested by those used in Egypt, or in Eastern Persia, or in China, or on the plains of Mesopotamia. We are quite in the dark as to these early steps; but as to their development in India, the approximate period of the rise of their essential feature of place value, their introduction into the Arab civilization, and their spread to the West, we have more or less definite information. When, therefore, we consider the rise of the numerals in the land of the Sindhu,^[51] it must be understood that it is only the large movement that is meant, and that there must further be considered the numerous possible sources outside of India itself and long anterior to the first prominent appearance of the number symbols.

No one attempts to examine any detail in the history of ancient India without being struck with the great dearth of reliable material.^[52] So little sympathy have the people with any save those of their own caste that a general literature is wholly lacking, and it is only in the observations of strangers that any all-round view of scientific progress is to be found. There is evidence that primary schools existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized.^[53] In the Vedic period, say from 2000 to 1400 B.C., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt, a fact attested by the Vedas themselves.^[54] Such advance in science presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant and probably always shall be. One of the Buddhist sacred books, the Lalitavistara, relates that when the Bōdhisattva^[55] was of age to marry, the father of Gopa, his intended bride, demanded an examination of the five hundred suitors, the subjects including arithmetic, writing, the lute, and archery. Having vanquished his rivals in all else, he is matched against Arjuna the great arithmetician and is asked to express numbers greater than 100 kotis.^[56] In reply he gave a scheme of number names as high as 10⁵³, adding that he could proceed as far as 10⁴²¹,^[57] all of which suggests the system of Archimedes and the unsettled question of the indebtedness of the West to the East in the realm of ancient mathematics.^[58] Sir Edwin Arnold, in The Light of Asia, does not mention this part of the contest, but he speaks of Buddha's training at the hands of the learned Viṣvamitra:

Thereupon Viṣvamitra Ācārya^[61] expresses his approval of the task, and asks to hear the "measure of the line" as far as yōjana, the longest measure bearing name. This given, Buddha adds:

It is needless to say that this is far from being history. And yet it puts in charming rhythm only what the ancient Lalitavistara relates of the number-series of the Buddha's time. While it extends beyond all reason, nevertheless it reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement.

To this pre-Christian period belong also the Vedāṅgas, or "limbs for supporting the Veda," part of that great branch of Hindu literature known as Smṛiti (recollection), that which was to be handed down by tradition. Of these the sixth is known as Jyotiṣa (astronomy), a short treatise of only thirty-six verses, written not earlier than 300 B.C., and affording us some knowledge of the extent of number work in that period.^[62] The Hindus also speak of eighteen ancient Siddhāntas or astronomical works, which, though mostly lost, confirm this evidence.^[63]

As to authentic histories, however, there exist in India none relating to the period before the Mohammedan era (622 A.D.). About all that we know of the earlier civilization is what we glean from the two great epics, the Mahābhārata^[64] and the Rāmāyana, from coins, and from a few inscriptions.^[65]

It is with this unsatisfactory material, then, that we have to deal in searching for the early history of the Hindu-Arabic numerals, and the fact that many unsolved problems exist and will continue to exist is no longer strange when we consider the conditions. It is rather surprising that so much has been discovered within a century, than that we are so uncertain as to origins and dates and the early spread of the system. The probability being that writing was not introduced into India before the close of the fourth century B.C., and literature existing only in spoken form prior to that period,^[66] the number work was doubtless that of all primitive peoples, palpable, merely a matter of placing sticks or cowries or pebbles on the ground, of marking a sand-covered board, or of cutting notches or tying cords as is still done in parts of Southern India to-day.^[67]

The early Hindu numerals^[68] may be classified into three great groups, (1) the Kharoṣṭhī, (2) the Brāhmī, and (3) the word and letter forms; and these will be considered in order.

The Kharoṣṭhī numerals are found in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandhāra, now eastern Afghanistan and northern Punjab. The alphabet of the language is found in inscriptions dating from the fourth century B.C. to the third century A.D., and from the fact that the words are written from right to left it is assumed to be of Semitic origin. No numerals, however, have been found in the earliest of these inscriptions, number-names probably having been written out in words as was the custom with many ancient peoples. Not until the time of the powerful King Aśoka, in the third century B.C., do numerals appear in any inscriptions thus far discovered; and then only in the primitive form of marks, quite as they would be found in Egypt, Greece, Rome, or in various other parts of the world. These Aśoka^[69] inscriptions, some thirty in all, are found in widely separated parts of India, often on columns, and are in the various vernaculars that were familiar to the people. Two are in the Kharoṣṭhī characters, and the rest in some form of Brāhmī. In the Kharoṣṭhī inscriptions only four numerals have been found, and these are merely vertical marks for one, two, four, and five, thus: